Optimal. Leaf size=124 \[ \frac{x \left (9 c d^2-e (5 b d-a e)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3} \]
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Rubi [A] time = 0.137904, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1257, 1157, 388, 205} \[ \frac{x \left (9 c d^2-e (5 b d-a e)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3} \]
Antiderivative was successfully verified.
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Rule 1257
Rule 1157
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac{\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}-\frac{\int \frac{-c d^2+b d e-a e^2+4 e (c d-b e) x^2-4 c e^2 x^4}{\left (d+e x^2\right )^2} \, dx}{4 e^3}\\ &=-\frac{\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac{\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}+\frac{\int \frac{-7 c d^2+e (3 b d+a e)+8 c d e x^2}{d+e x^2} \, dx}{8 d e^3}\\ &=\frac{c x}{e^3}-\frac{\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac{\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}-\frac{\left (15 c d^2-e (3 b d+a e)\right ) \int \frac{1}{d+e x^2} \, dx}{8 d e^3}\\ &=\frac{c x}{e^3}-\frac{\left (c d^2-b d e+a e^2\right ) x}{4 e^3 \left (d+e x^2\right )^2}+\frac{\left (9 c d^2-e (5 b d-a e)\right ) x}{8 d e^3 \left (d+e x^2\right )}-\frac{\left (15 c d^2-e (3 b d+a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.10437, size = 122, normalized size = 0.98 \[ \frac{x \left (a e^2-5 b d e+9 c d^2\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-a e^2-3 b d e+15 c d^2\right )}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 179, normalized size = 1.4 \begin{align*}{\frac{cx}{{e}^{3}}}+{\frac{{x}^{3}a}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{5\,{x}^{3}b}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{9\,{x}^{3}cd}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{ax}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,bdx}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,c{d}^{2}x}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{a}{8\,de}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,b}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,cd}{8\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7991, size = 873, normalized size = 7.04 \begin{align*} \left [\frac{16 \, c d^{2} e^{3} x^{5} + 2 \,{\left (25 \, c d^{3} e^{2} - 5 \, b d^{2} e^{3} + a d e^{4}\right )} x^{3} +{\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} +{\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \,{\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) + 2 \,{\left (15 \, c d^{4} e - 3 \, b d^{3} e^{2} - a d^{2} e^{3}\right )} x}{16 \,{\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )}}, \frac{8 \, c d^{2} e^{3} x^{5} +{\left (25 \, c d^{3} e^{2} - 5 \, b d^{2} e^{3} + a d e^{4}\right )} x^{3} -{\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} +{\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \,{\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (15 \, c d^{4} e - 3 \, b d^{3} e^{2} - a d^{2} e^{3}\right )} x}{8 \,{\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.09714, size = 201, normalized size = 1.62 \begin{align*} \frac{c x}{e^{3}} - \frac{\sqrt{- \frac{1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log{\left (- d^{2} e^{3} \sqrt{- \frac{1}{d^{3} e^{7}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log{\left (d^{2} e^{3} \sqrt{- \frac{1}{d^{3} e^{7}}} + x \right )}}{16} + \frac{x^{3} \left (a e^{3} - 5 b d e^{2} + 9 c d^{2} e\right ) + x \left (- a d e^{2} - 3 b d^{2} e + 7 c d^{3}\right )}{8 d^{3} e^{3} + 16 d^{2} e^{4} x^{2} + 8 d e^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08884, size = 144, normalized size = 1.16 \begin{align*} c x e^{\left (-3\right )} - \frac{{\left (15 \, c d^{2} - 3 \, b d e - a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{8 \, d^{\frac{3}{2}}} + \frac{{\left (9 \, c d^{2} x^{3} e - 5 \, b d x^{3} e^{2} + 7 \, c d^{3} x + a x^{3} e^{3} - 3 \, b d^{2} x e - a d x e^{2}\right )} e^{\left (-3\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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